Fall 2001 16.3114-1 Reference Inputs So far we have looked at how to pick K to get the dynamics to have some nice properties(i.e stabilize A) The question remains as to how well this controller allows us to track a reference command Performance issue rather than just stability ● Started with A z+ Bu Ka For good tracking performance we want y(t)≈r(t)ast→∞ Consider this performance issue in the frequency domain. Use the final value theorem lim g(t)=lim sY(s) t→→∞o S→0 Thus, for good performance, we want sY(s)≈sR(s)ass→0→ RO So, for good performance, the transfer function from R(s)to Y(s should be approximately l at dcFall 2001 16.31 14–1 Reference Inputs • So far we have looked at how to pick K to get the dynamics to have some nice properties (i.e. stabilize A) • The question remains as to how well this controller allows us to track a reference command? – Performance issue rather than just stability. • Started with x˙ = Ax + Bu y = Cx u = r − Kx • For good tracking performance we want y(t) ≈ r(t) as t → ∞ • Consider this performance issue in the frequency domain. Use the final value theorem: lim t→∞ y(t) = lim s→0 sY (s) Thus, for good performance, we want sY (s) ≈ sR(s) as s → 0 ⇒ Y (s) R(s)



s=0 = 1 • So, for good performance, the transfer function from R(s) to Y (s) should be approximately 1 at DC